Double perfect partitions

MacMahon [Combinatory Analysis, vols. I and II, Cambridge University Press, Cambridge, 1915, 1916 (reprinted, Chelsea, 1960)] introduced a perfect partition of positive integer n, which is a partition such that every positive integer less than or equal to n can be uniquely represented by the sum of its parts. We generalize perfect partition and find a relation with ordered factorizations.     

New interpretations for noncrossing partitions of classical types

We interpret noncrossing partitions of type B and type D in terms of noncrossing partitions of type A. As an application, we get type-preserving bijections between noncrossing and nonnesting partitions of type B, type C and type D which are different from those in the recent work of Fink and Giraldo. We also define Catalan tableaux of type B and type D, and find bijections between them and noncrossing partitions of type B and type D respectively.     

A shuffling theorem for reflectively symmetric tilings

The author and Rohatgi recently proved a ‘shuffling theorem’ for doubly-dented hexagons. In particular, they showed that shuffling removed unit triangles along a horizontal axis in a hexagon changes the tiling number by only a simple multiplicative factor. In this paper, we consider a similar phenomenon for a symmetry class of tilings, namely, the reflectively symmetric tilings. We also prove several shuffling theorems for halved hexagons.     

Analysis of some new partition statistics

We study several statistics for integer partitions: for a random partition of an integer n we consider the average size of the smallest gap (missing part size), the multiplicity of the largest part, and the largest repeated part size. Furthermore, we estimate the number of gap-free partitions of n. 2000 Mathematics Subject Classification Primary—05A17; Secondary—11P82 Dedicated to Helmut Prodinger on the occasion of his 50th birthday P.J. Grabner is supported by the START-project Y96-MAT of the Austrian Science Fund. This material is based upon work supported by the National Research Foundation under grant number 2053740.     

Formulas for zonal polynomials

New integral and differential formulas for zonal polynomials are proved. As illustrations, zonal polynomials corresponding to partitions of two parts are computed. A method is presented, based on a certain partial differential operator, for expressing an orthogonally invariant polynomial as a linear combination of zonals. Zonal polynomials are expressed as linear combinations of well-known symmetric polynomials.     

Efficient multipole expansion: Choice of order and density partitioning techniques

Two approaches to improve the convergence of the multipole series were considered: 1) an increase in the order of the expansion; 2) decomposition of the molecular charge density into smaller distributions. New decompositions of the molecular electronic density and a computational procedure to generate high-order moments are presented. The accuracy and timing of test calculations on the H₂O ... H₂O system are given and suggestions are made for optimizing the choice of an expansion for more general systems.We express our appreciation to the National Institutes of Health which supported this work under Grant 1 R01 GM 20436-02.     

The <Emphasis Type="Italic">q</Emphasis>-Variations of Sylvester’s Bijection Between Odd and Strict Partitions

The q-identities corresponding to Sylvester’s bijection between odd and strict partitions are investigated. In particular, we show that Sylvester’s bijection implies the Rogers-Fine identity and give a simple proof of a partition theorem of Fine, which does not follow directly from Sylvester’s bijection. Finally, the so-called (m, c)-analogues of Sylvester’s bijection are also discussed.2000 Mathematics Subject Classification: Primary—05A17, 05A15, 33D15, 11P83     

Jagged Partitions

By jagged partitions we refer to an ordered collection of non-negative integers (n₁, n₂,..., n_m) with n_m≥ p for some positive integer p, further subject to some weakly decreasing conditions that prevent them for being genuine partitions. The case analyzed in greater detail here corresponds to p = 1 and the following conditions n_i≥ n_i+1−1 and n_i≥ n_i+2. A number of properties for the corresponding partition function are derived, including rather remarkable congruence relations. An interesting application of jagged partitions concerns the derivation of generating functions for enumerating partitions with special restrictions, a point that is illustrated with various examples. 2000 Mathematics Subject Classification: Primary—05A15, 05A17, 05A19     

An Application of Polya's Enumeration Theorem to Partitions of Subsets of Positive Integers

Let S be a non-empty subset of positive integers. A partition of a positive integer n into S is a finite nondecreasing sequence of positive integers a ₁, a ₂,...,a _r in S with repetitions allowed such that . Here we apply Polya's enumeration theorem to find the number P(n; S) of partitions of n into S, and the number DP(n; S) of distinct partitions of n into S. We also present recursive formulas for computing P(n; S) and DP(n; S).